Re: [xmca] Further thoughts on Sfard's and Davydov's approaches to learning

[David Kellogg <vaughndogblack@yahoo.com>]

Andy, Larry (Anna? David Kirshner? Are you out there?):
 
Jean Schmittau is the name you want. She tried Davydov's method on some New York elementary school students, and obtained some really amazing results. But the key difference turned out to be not between "activity" and "discourse" but rather between ways of teaching that posed the question of "part-whole" relationships. Measuring turned out to be rather better at this than counting, that's all.
 
We have been working on a slightly different problem. It's this:
 
T:How many apples?
S1:One.
S2:Two.
T: More than one? Less than two?
S1: More than one!
S2: Less than two!
T: Good. Now we are getting somewhere. But how MUCH more? How MUCH less?
 
You can see that counting and measuring are really connected here. But what are they connected TO?
 
Ss…"ban" (Half!)
T: Yes, “half”. But you know, “half” sometimes just means “a chunk” or “a hunk”. That’s not  good enough for math class, is it?
Ss: No!
T: Is it good enough for science class?
Ss: No!
T: Is it good enough for lunchtime?
Ss: Yes! No!
T: Hmmm…let’s be a little more mathematical. A little more scientific. Let’s call it something else. One plus two? No, that’s too much. Two minus one? Too much or too little?
Ss: Too little!
T: Right. Let’s see…suppose we take S1’s answer. And we DIVIDE it by S2’s answer. Like this!
Ss: 1/2!
T: Good! One…divided by…two. Or two divided by one? Which is bigger?
Ss: One divided by two! Two divided by one!
T: OK--two divided by one is too big. So let’s take one divided by two. Now, suppose we take S1’s answer. And we ADD this. What do we get?
Ss: One and half. Three halves. One point five.
T: Right. Now, suppose we take S2’s answer. Should we ADD or SUBTRACT this?
 …
 
Three things to notice here. 
 
a) First of all, the answer is created by UPTAKING both answers and relating them in some way. 
 
b) Secondly, the answer is created in a GENERALIZEABLE (not a lexical) way--by a relation, an operation, which can be generalized.
 
c) Thirdly, the answer is not complete. The teacher NOW has to play a game to demonstrate how it can be generalized into a system. 
 
The game works like this. We start with a number, e.g. 3/2. Each child has to "rename" the number (e.g. 6/4, 1.5, etc.) and when a child makes a mistake, she sits down. Last "man" standing wins (one and a half apples is the prize, but they have to be shared!)
 
Now, for this to reveal the basic truth that Vygotsky hammers home in Chapter Six (viz. that ALL numbers, and indeed ALL CONCEPTS can be renamed an infinite number of ways, as long as they belong to a system), there are TWO things that have to be generalized, and they are logically separate:
 
First of all, it has to be seen that you can start with ANY number. This cannot be directly taught (because if the teacher suggests a number, it is no longer any number) but the teacher can be carefully careless about it, like this:
 
T: First, I give a number. Let's say...3/2. Nah, that's boring. We did three halves. Has anyone got a more interesting number?
 
Secondly, it has to be seen that the game can go on FOREVER. Again, this cannot be directly taught (because whenever a child makes a mistake the series ends). It can only happen as a result of play. 
 
I guess what I would say about this is that it is not really about discourse or activity, but rather about play. Of course, both discourse and activity CAN be seen as play and indeed ARE seen as play by children. But the key component is not so much mediation as contradiction.
 
S2 contradicts S1. The teacher uptakes their contradiction and relates them in some way. 
 
Division contradicts addition and multiplication, and allows "two" to contradict "one". The teacher uptakes this contradiction and relates it in an operation that can be generalized. 
 
In generalization, the general contradicts the particular by making it universal and no longer particular. The teacher does this by playing a game in which "one" can be "anyone" and a finite series is potentially an infinite one (see "Any and Infinite").
 
But so do the past, the present, and Walt Whitman himself:
 
The past and present wilt - I have fill'd them, emptied them. 
And proceed to fill my next fold of the future. 
 
Do I contradict myself? 
Very well then I contradict myself, 
(I am large, I contain multitudes.) 
 
(Walt Whitman, Song of Myself)
 
David Kellogg
Seoul National University of Education

--- On Thu, 6/30/11, Andy Blunden <ablunden@mira.net> wrote:


From: Andy Blunden <ablunden@mira.net>
Subject: Re: [xmca] Further thoughts on Sfard's and Davydov's approaches to learning
To: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>
Date: Thursday, June 30, 2011, 10:02 PM


Larry, I guess that both counting and measuring can be starting points for good teaching of mathematics (but I'd really like to hear from a maths teacher about that!) and I don't see any barrier to an eclectic approach either. I suppose I am interested in the relation between Activity and Discourse which is implicit in this problem. The problem posed by Doyle, that gaps in teaching may be being solved by things happening outside the classroom is intriguing as well. If kids are measuring and/or counting in everyday life, does this make the choice of how they are introduced to mathematical formalism in the classroom moot? Or is it the whole point? Does this issue have anything to do with the idea that practical intelligence is developmentally prior to word-use? Is there a basis for seeing Discourse as a special case of Activity here? Is it possible to found a philosophical viewpoint on Discourse without an implicit and unstated foundation in Activity? And
 how would this show up in problems of learning maths? Does basing maths teaching in Activity simply postpone the real work until you get up to mathematical discourse? Is there are class difference? (Discourse for the class of symbolic analysts, Activity for the class of manual workers?) Do the same issues arise in teaching Literature?

Anyone home?

Andy

Larry Purss wrote:
> Andy, some further thoughts on the topic you opened up on contrasting Anna's
> and Davydov's approaches to learning.
> I was not able to download Anna's article because of the way its formatted.
> However I went to Anna's website and have downloaded two of her articles and
> also the introductory chapter of her book written in 2008. [The first half
> of this book is elaborating her theoretical position that thinking IS
> communication, as particular forms of discourse]
> She says her approach is similar to Harre's approach to "discursive
> psychology" and both Anna and Harre definitely view thinking in a dialogical
> way as a particular form of conversation with one's self.  Anna sees this
> communicative perspective on thinking as learning particular "forms of
> discourse" which have developed historically and now children must learn
> these particular discourse procedures through entering mathematical
> conversations as forms of social interaction in order to learn to think
> "mathematically".
> 
> Andy, my understanding of Davydov and Gal'perin is that once  "systems of
> discourse" have developed historically as "systems of meaning" it is far
> more efficient to start from the very beginning to introduce the entire
> system and not build up to the system perspective FROM the more concrete
> procedures such as counting objects.  Therefore measurement [as
> fundamentally relational] is prior to counting.
> 
> If I've got these basic premises of Anna's and Davydov's procedures
> accurate, do you see these procedural approaches as complimentary or are
> they challenging each others basic assumptions.   If they are pointing to
> different procedures and assumptions of the best way to approach learning,
> then definitely this is a topic to tease out assumptions about  fundamental
> concepts of learning?
> 
> Larry
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>   

-- ------------------------------------------------------------------------
*Andy Blunden*
Joint Editor MCA: http://www.informaworld.com/smpp/title~db=all~content=g932564744
Home Page: http://home.mira.net/~andy/
Book: http://www.brill.nl/default.aspx?partid=227&pid=34857
MIA: http://www.marxists.org

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